Lindblad equation

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In quantum mechanics, the Lindblad equation or master equation in the Lindblad form is the most general type of markovian master equation describing non-unitary (dissipative) evolution of the density matrix ρ that is trace preserving and completely positive for any initial condition. Non-Markovian processes can produce Markovian master equations, but they will only preserve positivity (and not complete positivity) and thus they will not be of Lindblad form. Complete positivity allows us to mix and match Lindblad terms and system Hamiltonians without breaking positivity.

The Lindblad master equation can be written:

\dot\rho=-{i\over\hbar}[H,\rho]-{1\over\hbar}\sum_{n,m}h_{n,m}\big(\rho L_m L_n+L_m L_n\rho-2L_n\rho L_m\Big)+\mathrm{h.c.}

where \ \rho is the density matrix, \ H is the Hamiltonian part, \ L_m are operators defined to model the system as are the constants \ h_{n,m}. The abbreviation "h.c." stands for hermitian conjugate. If the \ L terms are all zero, then this is the quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem.

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has \ L_0=a, \ L_1=a^{\dagger}, \ h_{0,1}=-(\gamma/2)(\bar n+1), \ h_{1,0}=-(\gamma/2)\bar n with all others \ h_{n,m}=0. Here \bar n is the mean number of excitations in the reservoir damping the oscillator and \ \gamma is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods.

[edit] References

  • Lindblad G, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48 119 (1976)
  • Gorini V, Kossakowski A and Sudarshan E C G, Completely positive semigroups of N-level systems J. Math. Phys. 17 821 (1976)
  • Banks T, Susskind L, and Peskin M E, Difficulties for the evolution of pure states into mixed states, Nuclear Physics B 244 (1984) 125-134
  • C. W. Gardiner and Peter Zoller, Quantum Noise, Springer-Verlag (1991, 2000, 2004).